Properties

Label 4-1305e2-1.1-c0e2-0-1
Degree $4$
Conductor $1703025$
Sign $1$
Analytic cond. $0.424165$
Root an. cond. $0.807019$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·11-s − 16-s − 2·19-s − 25-s + 2·31-s − 2·41-s + 2·49-s + 2·61-s − 2·79-s + 2·89-s + 2·101-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 2·176-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 2·11-s − 16-s − 2·19-s − 25-s + 2·31-s − 2·41-s + 2·49-s + 2·61-s − 2·79-s + 2·89-s + 2·101-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s − 2·176-s + 179-s + 181-s + 191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1703025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1703025\)    =    \(3^{4} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(0.424165\)
Root analytic conductor: \(0.807019\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1703025,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096909928\)
\(L(\frac12)\) \(\approx\) \(1.096909928\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3 \( 1 \)
5$C_2$ \( 1 + T^{2} \)
29$C_2$ \( 1 + T^{2} \)
good2$C_2^2$ \( 1 + T^{4} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
17$C_2^2$ \( 1 + T^{4} \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
43$C_2^2$ \( 1 + T^{4} \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
59$C_2$ \( ( 1 + T^{2} )^{2} \)
61$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
97$C_2^2$ \( 1 + T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.07534037044711140697249086762, −9.629581589835499884178574154188, −9.216086368502800947316544361078, −8.708975881360725939163288443820, −8.425781577045465267046547805952, −8.398535032696022107810252424803, −7.51606806601478066993358413024, −6.99929281343331690277211431559, −6.80499541869425218127789404476, −6.33704967617512391372602249254, −6.08288116735860942898724874131, −5.56878428209454045987033988496, −4.71304070565532912844897157729, −4.53237967084903742862860794444, −4.01098257948622441363279535042, −3.70845573790916081528985430577, −2.99664042348209914750916686386, −2.08776839966650407754470286570, −2.00937293161635186341752181071, −0.940804633132667357714633638086, 0.940804633132667357714633638086, 2.00937293161635186341752181071, 2.08776839966650407754470286570, 2.99664042348209914750916686386, 3.70845573790916081528985430577, 4.01098257948622441363279535042, 4.53237967084903742862860794444, 4.71304070565532912844897157729, 5.56878428209454045987033988496, 6.08288116735860942898724874131, 6.33704967617512391372602249254, 6.80499541869425218127789404476, 6.99929281343331690277211431559, 7.51606806601478066993358413024, 8.398535032696022107810252424803, 8.425781577045465267046547805952, 8.708975881360725939163288443820, 9.216086368502800947316544361078, 9.629581589835499884178574154188, 10.07534037044711140697249086762

Graph of the $Z$-function along the critical line